Sol'n hwk 3 and 4:

With equidistant nodes, the lower bound is: 1.13272
                        the upper bound is: 1.61349

There are many choices for pblm #4, depending on the nodes.

I have done something that goes far beyond Calc 2, namely I have
determined the choices of nodes that give the best (smallest) upper
bound and the best (largest) lower bound that is possible with 10
intervals. This job requires to solve a minimax problem, not
with one variable (as in calc 1), but with nine variables, and I did it
numerically. (Not checking the ``second derivative'' however as I ought to
have done...)

Here are the results:

The best upper bound is: 1.55743
obtained with the nodes (rounded to 3 decimals) 
0.420, 0.667, 0.914, 1.185, 1.501, 1.882, 2.360, 2.983, 3.822
These nodes give the (good but not best) lower bound 1.19107

The best lower bound is: 1.20027
obtained with the nodes (rounded to 3 decimals) 
0.278, 0.507, 0.736, 0.987, 1.279, 1.636, 2.095, 2.717, 3.611
These nodes give the (good but not best) upper bound 1.56478


The actual value of the integral (also obtained by means that are yet 
ahead in class) is 1.3734


There are certainly examples where the effect of choosing good nodes is
more pronounced; here I selected a rather simple example just for the 
principle.